Problem 10 - Calculus for Biology and Medicine

Question

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) are independent random variables with \(P\left(X_{i}>x\right)=e^{-2 x} .\) What can you say about \(\frac{1}{n} \sum_{i=1}^{n} X_{i}\) as \(n \rightarrow \infty ?\)

Step-by-Step Solution

Answer

As \(n \rightarrow \infty\), \(\frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \frac{1}{2}\).

Step 1: Identify the Distribution

We see that \(P(X_i > x) = e^{-2x}\). This implies that each random variable \(X_i\) is exponentially distributed with a rate parameter \(\lambda = 2\). This is because the survival function of an exponential distribution with rate \(\lambda\) is \(e^{-\lambda x}\).

Step 2: Find Expectation of Each \(X_i\)

For an exponentially distributed random variable with rate \(\lambda\), the expected value is given by \(E[X_i] = \frac{1}{\lambda}\). Since \(\lambda = 2\), we have \(E[X_i] = \frac{1}{2}\).

Step 3: Calculate the Sum of Expectations

Using the linearity of expectation, the expected value of \(\frac{1}{n} \sum_{i=1}^{n} X_i\) is \(\frac{1}{n} \sum_{i=1}^{n} E[X_i] = \frac{1}{n} \times n \times \frac{1}{2} = \frac{1}{2}\).

Step 4: Apply the Law of Large Numbers (LLN)

According to the Law of Large Numbers, for independent and identically distributed random variables, the sample average converges to the expected value as the sample size \(n\) goes to infinity. Therefore, \(\frac{1}{n} \sum_{i=1}^{n} X_i \rightarrow \frac{1}{2}\) as \(n \rightarrow \infty\).

Key Concepts

Exponential Distribution

The Exponential Distribution is vital in understanding a range of phenomena, particularly in the context of time-based events. It describes the time between events in a Poisson process, which are events that occur continuously and independently at a constant average rate. In our exercise, we saw that each random variable \( X_i \) is exponentially distributed with a rate parameter \( \lambda = 2 \). The rate parameter indicates how often we expect the events to happen.

• The survival function, or the probability that \( X_i \) exceeds \( x \), is \( P(X_i > x) = e^{-\lambda x} \). This measure expresses how long it takes for an event to occur.
• When \( \lambda = 2 \), it implies that the average waiting time between events is short, with faster occurrences.

The Exponential Distribution is memoryless, meaning the probability distribution of the time until the next event does not depend on how much time has already elapsed. This property simplifies many calculations in probability, making it a popular choice for modeling."

Expectation

Expectation, often referred to as the expected value or mean, is a fundamental concept in probability and statistics. It provides the average outcome that we would anticipate after many trials of a random process.

• The expectation of an exponentially distributed random variable \( X \) with rate \( \lambda \) is \( E[X] = \frac{1}{\lambda} \). In simpler terms, it describes the typical or average value we expect from our random variable.
• For our case, since \( \lambda = 2 \), the expected value \( E[X] = \frac{1}{2} \). This tells us that, on average, we expect each random variable \( X_i \) to be \( 0.5 \) as we perform repeated sampling.

The expectation is crucial when discussing the Law of Large Numbers, as it represents the value to which the sample average converges with increasing trials."

Convergence

Convergence is an essential concept in probability and statistics, particularly regarding the behavior of sequences of random variables. The Law of Large Numbers (LLN) is a classic theorem that deals with convergence.

• The LLN states that for a sequence of independent and identically distributed random variables, the average of these variables converges in probability to the expected value as the number of variables \( n \rightarrow \infty \).
• In our exercise, as \( n \) approaches infinity, the sample average \( \frac{1}{n} \sum_{i=1}^{n} X_i \) converges to \( \frac{1}{2} \). This signifies that the larger the sample size, the closer the sample mean gets to the population mean, ensuring accuracy and reliability in predictions.

Understanding convergence helps in assessing the long-term average behavior of stochastic processes and confirms the reliability of empirical averages to approximate theoretical means."